Abstract

We design novel photonic crystal slab heterostructures, substituting the air in the holes with materials of refractive index higher than n=1. This can be achieved by infiltrating the photonic crystal slab (PCS) with liquid crystal, polymer or nano-porous silica. We find that the heterostructures designed in this way can have quality factors up to Q=10 6. This high-Q result is comparable with the result of previously reported designs in which the lattice is elongated in one direction. Unlike conventional heterostructures, our design does not require nanometre-scale changes in the geometry. Additionally, infiltrated PCS can be constructed at any time after PCS fabrication.

A cavity is usually formed in either of two ways: forming a point cavity or forming a “hetero-structure”. A point defect may be formed by omitting one or more holes in the centre of the slab. In that case the optimization of the optical nanocavity design, i.e. a further increase of a quality factor, is possible by modifying the geometry of the lattice structure surrounding the cavity [11–13

]. These lattices are combined similarly to the design illustrated in Fig. 1(b) and (c): in the outer regions (PC1) the lattice is hexagonal, whereas in the central region (PC2), the lattice is slightly elongated (by 10 nm) in the direction orthogonal to the heterostructure (parallel to the waveguide), but is otherwise identical. A waveguide introduced across these PCSs has different dispersion curves within the different parts of the PCS [15

]. Therefore within the same photonic band gap (PBG) there is a “mode-gap” between these curves. The mode “propagates” in the waveguide of the central structure and decays exponentially elsewhere. Consequently the light in one of the photonic crystal waveguides can be localized due to the mode-gap effect. With this design, Song et al experimentally achieved a quality factor of Q=6×105 [14

]. In this way Kuramochi et al achieved experimentally a quality factor of Q=8×105.

Current methods to realize high-Q cavities, in both point and heterostructures cavity design, typically rely on extremely precise control of holes’ size and position through nanolithographic techniques such as electron-beam lithography. Because the design requires high precision, the positioning of the holes may be a limiting factor in achieving high-Q cavities, as pointed out by Song et al in Ref [17

]. Whilst the previous designs rely on small lattice constant perturbations or hole shifts of orders of nanometers, both of which are hard to achieve precisely, here we propose a design that can be fabricated without any change in the geometry of the regular structure. In this paper, we use novel heterostructure designs changing the refractive index within the holes in the central part of the homogenous structure. For example the air in the holes of PC2 can be replaced with material of refractive index n>1. Here we consider materials having refractive index in the range n=1.1-1.7. The aim of this design is to increase the average refractive index of the PCS which, to lowest order, has the effect of lowering the optical frequency of features in the photonic band structure of PC2 with respect to that of PC1. It has been experimentally demonstrated that a nematic liquid crystal (LC) infiltrated photonic crystal laser can be constructed by encasing the PCS between two indium tin oxide glass plates [9

]. For design purposes in this paper we take the LC refractive index to be isotropic. For future applications it may be necessary to take into account the tensor nature of the LC optical properties. The refractive index range between air and polymers matches porous silica. Different properties of sol-gel derived nano-porous silica films within the refractive index range n=1.14-1.25 can be obtained by a mixed atmosphere treatment [21

2. Model and method

Our model is a PCS composed of a hexagonal array of cylindrical air holes in a silicon slab, as illustrated in Fig. 1. The structure has holes of radius R, a is the lattice constant and h is the thickness of the slab. Across the PCS there is a line defect, a W1 waveguide, in the Γ-K direction. We start with a homogeneous PCS, as illustrated in Fig. 1(a) and design the heterostructures by changing the holes’ refractive index in the central region of the PCS (indicated by the darker circles in Fig. 1(b) and (c)).

First we consider a bulk silicon-based (n=3.4) PCS that is infinite in the plane in order to obtain PBGs and associated eigenstates of a waveguide introduced in the Γ-K direction. As the second step in the design, a finite PCS, with 25a in the x-direction and 25a in the y-direction, is considered with the cavity in the centre.

Fig. 1 (a) Schematic of PCS with a W1 waveguide in the Γ-K direction and refractive index distribution in the plane of the structures considered (b) m=1 and (c) m=4.

For both structures the hole radius is R=0.29a and the thickness of the slab is h=0.6a. We start our analysis with the cavity illustrated in Fig. 1(b) because it is an analogue to the structure described by Song et al in terms of the cavity length [14

]. Next we consider structures that have longer cavities such that illustrated in Fig. 1(c). The PC2 length is denoted by d, d=ma+2R, where m is integer.

In order to design a cavity, two numerical methods are used: the plane wave expansion (PWE) method for the PBG calculations and associated eigenstates of the photonic crystal waveguide, and the finite-difference time-domain (FDTD) method, combined with techniques of fast harmonic analysis [22

] for the quality factor calculations. This method exploits the knowledge that for a signal consisting of one or a few resonant modes, the electric field at an arbitrary point as a function of time can be represented as a sum of complex exponentials. By projecting the signal onto a Fourier basis in a narrow range around the resonant frequency, the complex frequencies can be found to very high accuracy, much greater than would be extracted from a standard Fourier transform. The error in the complex frequency is dominated entirely by the spatial grid resolution rather than the length of the simulation. Details on numerical parameters for the calculations can be found in Ref [5

]. Therefore we first examine if there is a sufficient mode-gap between structures having materials other than air within the holes. In Fig. 2(a) we plot the dispersion curves for the regular structure (PC1) and modified structure (PC2).

Both structures have two guided modes below the light line in the lowest PBG, one in the middle of the bandgap and the other one in the lower part of the bandgap. The lower mode is the mode of interest [18

]. The dispersion curves of this mode for both regular structure, PC1, and the PC2 where air holes are filled with material having refractive index n=1.6, are plotted in Fig. 2(a). In the same figure the lower band edge is indicated by solid horizontal lines both for the regular and modified structure. Obviously, filling the holes with material of higher refractive index than air increases the refractive index of the structure in whole and consequently lowers the dispersion curve. The gap between these dispersion curves, measured at the edge of the Brillouin zone, is Δω˜=3.25×10-3, where ω˜=ωa/2πc. The size of the mode-gap is comparable with the mode-gap of the heterostructures formed of different lattice constants PCSs [10

]. This indicates that the heterostructures formed of the PCSs that differ in the holes’ refractive index are also capable of the mode-gap operation.

Fig. 2 (a) Dispersion curves for W11 within the region of the lowest gap of the regular structure PC1 (empty triangles) and W12 of the structure PC2 (full circles), nholes=1.6; the dashed line represents the light line, the horizontal solid lines represent the lower band gap edge for PC1 and PC2, (b) quality factor Q (rectangles) and modal volume V (crosses) as a function of the refractive index of the central holes for m=1.

Now we calculate the quality factors for the structures shown in Fig. 1(b). The structures are similar to the structures described by Song et al in Ref [14

]. The refractive index of the holes in PC2 is varied between n=1.1 and n=1.7. The results for the quality factors and modal volumes of the resonant modes are plotted in Fig. 2(b). The maximum quality factor of Q=2.5×105 appears at n=1.4. As the holes’ refractive index is increased, the average refractive index of the structures increases. This results in better out-of-plane confinement and therefore smaller out-of-plane losses, increasing the Q. However at n=1.4 the Q starts to decrease. We believe this happens because the dispersion curves for higher refractive indices shift lower whilst the lower band edge for PC1, denoted by the upper horizontal solid line in Fig. 2(a), is fixed. Consequently with increasing index, the dispersion curve of PC2 approaches the lower band gap edge of PC1.

It is worth pointing out that there is a large range of refractive indices, n=1.25-1.6, where the quality factors are of order of 105. This coincides with the refractive indices of polymer materials and liquid crystals [8–10

]. The results for modal volumes of these resonances, expressed in (λ/n)3 with n=3.4, are also plotted in Fig. 2(b). As the refractive index in the central holes increases the modal volume decreases from V=2.11 (λ/n)3 to V=1.17 (λ/n)3. This is expected behaviour as the resonant mode is becoming better confined with the increased difference between the two heterostructures. The ratios Q/V, Q2/V, important for many applications discussed in Section 1, still have their maximum at n=1.4, with the modal volume of V=1.46 (λ/n)3 at that refractive index.

Next we investigate the effect of the cavity length on the quality factor and modal volume. Filling more holes changes cavity length, an example is illustrated in Fig. 1(c). We calculate quality factors for the cavities d=ma+2R, where m=1,2,…,5. The results for the fixed refractive index n=1.4 are shown in Fig. 3(a). Up to m=4, increasing the length increases the quality factor with the maximum exceeding Q=6×105. The modal volume does not change significantly with m, varying by less than 10 %.

The resonant frequencies are also plotted in Fig. 3(a) and the mode-gap edges are indicated by the horizontal dotted lines. As the refractive index is fixed, the mode-gap that ranges from ω˜=0.2636 to ω˜=0.2607, does not change as m changes. The resonant frequency for m=1 occurs just below the upper mode-gap edge. As m increases the frequency crosses over the mode-gap almost linearly, passing the mid mode-gap closest to m=3. The resonant frequency that corresponds to the maximum, ω˜=0.2617, is in the lower half of the mode-gap. These results suggest that the relative position of the resonant frequency within the mode-gap is an important parameter in the design of high-Q heterostructures as well as the mode-gap position within the PBG as discussed above.

Fig. 3 Quality factor Q (rectangles) (a) and resonant frequencies (crosses) as a function of the number of periods within the cavity m, for fixed nholes=1.4, (b) as a function of the refractive index of the central holes for m=4 cavity.

As the maximum Q appears at m=4 we fix the cavity length at that value, see Fig. 1(c), and vary the holes’ refractive index in the range n=1.15-1.5. The results are shown in Fig. 3(b). The maximum value of Q=9.7×105 is achieved at n=1.25. In practice this structure can be attained by filling the holes with nano-porous silica [21

]. If polymer is used the quality factor decreases but still remains high. For example, filling the holes with polyhexafluoropropylene oxide with the refractive index of n=1.3 provides a high-Q cavity Q=7.6×105. The modal volume plotted in the same figure decreases as the holes’ refractive index increases as is the case for the cavity that consists of one period. The modal volume that corresponds to the maximum Q is V=1.56 (λ/n)3.

We compare these results with the results presented in Fig. 2(b). The maximum occurs at different refractive index values, for m=1 at n=1.4 and for m=4 at n=1.25. However there is no contradiction as the resonant frequencies are very close, for m=1, ω˜=0.2628 and for m=4, ω˜=0.2625. As expected, increasing the number of layers lowers the resonant frequency.

Note that a second family of symmetric heterostructures, which are shifted by half a period in the horizontal direction from those in Figs 1(b) and (c), can be constructed. However, the Q values of these are approximately an order of magnitude smaller then those presented here. For this reason, we do not consider them here in more details.

4. Discussion and Conclusions

The use of polymers and liquid crystals for a point cavity design decreases the quality factor because of the weaker vertical confinement that increases the out-of-plain losses [8–10

]. On the other hand we show here that filling the holes with these materials enables ultrahigh-Q heterostructures because it allows for the mode-gap operation that relies on the refractive index perturbation. An additional advantage of the heterostructures cavity design over the point cavity design is the large range of parameters that provide high-Q cavities. Even though the modal volume is slightly higher for heterostrucures than for the point cavities, this difference is minor compared to the large differences in Q.

We designed high-Q cavities with quality factors that are comparable those obtained for heterostructures with lattice variation [14

]. However there is no evidence that this design is less sensitive that the geometry based design. Susceptibility of the cavity properties to variations in the hole filling process will be addressed in future studies. The processing of air-hole infiltration can be done at any time after fabrication. If the structure is filled with LC, electro-optic or nonlinear polymer there is also the possibility of tuning these structures when voltage is applied. In future studies the effect of the top cladding layer produced during LC cell construction [10

In conclusion we have shown that ultrahigh-Q cavities can be designed in PCS heterostructures without change of the structure geometry. Quality factors of order Q~106 can be obtained by filling the holes in the central region of the homogenous PCS with nanoporous silica. The maximum values of this design achievable by using polymer materials or LC are higher than Q=7×105. This approach represents a novel technique for creating ultrahigh-Q cavities that furthermore opens the possibility of post-processing in PCS-based microcavities.

Acknowledgment

This work was produced with the assistance of the Australian Research Council (ARC) under the ARC Centres of Excellence Program. CUDOS (the Centre for Ultrahigh-bandwidth Devices for Optical Systems) is an ARC Centre of Excellence.

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